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2019issue C0648-55

Building a three-harmonic Fourier series cycle wave

A published Fourier analysis construction assembles a composite cycle wave from one Fundamental period, three second-order bandpass harmonics, and a relative-power mix, with an optional rate-of-change used in a coded next-bar rule.

  • The series model has one user input named Fundamental, with a default period of 20 bars, that sets the primary cycle length.
  • Filter coefficients are computed once from cosine terms at the fundamental period and at half and one-third of that period, using a fixed Bandwidth of 0.1.
  • Each harmonic is isolated with a second-order bandpass filter on close prices, paired with a quadrature series, then mixed by relative in-phase and quadrature power when fundamental power is nonzero.
  • An optional rate-of-change of the composite wave, scaled by the fundamental period divided by 12.57, is described as crossing zero at cyclic turning points and is the trigger in one coded next-bar market strategy.
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Current trackBandpass filter15 readingsFourier analysis15 readingsDominant cycle detection138 readings

A single fundamental period

The published Fourier analysis construction uses a single user input named Fundamental, with a default period of 20 bars, to set the primary cycle length for the series model. That Fundamental length is the dominant cycle detection setting used by every later stage.

Coefficients for three harmonics

Filter coefficients are computed once from cosine terms at the fundamental period and at half and one-third of that period, using a fixed Bandwidth of 0.1. Those three periods are the harmonics that the bandpass filter stages isolate.

Three-harmonic bandpass coefficients at a 20-bar fundamental

L, the center-frequency cosine, falls from 0.95 on the 20-bar fundamental to 0.59 on the third harmonic, which is how the three bandpass filters are tuned. G stays near 1 because bandwidth is only 0.1, and damping S eases from 0.97 to 0.91. These six-decimal figures are the precomputed L/G/S table on the Excel Fourier Market Model sheet, the coefficients a reconstruction of the composite cycle wave actually uses.
L, the center-frequency cosine, falls from 0.95 on the 20-bar fundamental to 0.59 on the third harmonic, which is how the three bandpass filters are tuned. G stays near 1 because bandwidth is only 0.1, and damping S eases from 0.97 to 0.91. These six-decimal figures are the precomputed L/G/S table on the Excel Fourier Market Model sheet, the coefficients a reconstruction of the composite cycle wave actually uses.

Computed once from Fundamental = 20 bars and Bandwidth = 0.1, then held fixed. Band-pass outputs are forced to zero for the first three bars and quadrature for the first four, so the composite wave is undefined at the start of a sample.

Bandpass isolation and a quadrature series

Each harmonic is isolated with a second-order bandpass filter on close prices. A quadrature series is then formed by scaling the first difference of that bandpass by the fundamental period divided by about 6.28.

Undefined history at the start

Bandpass outputs are forced to zero for the first three bars, and quadrature outputs are forced to zero for the first four bars, so the recursive filters do not start from undefined history.

Relative power and the composite wave

Relative harmonic amplitudes are estimated by summing in-phase and quadrature power over a lookback equal to the fundamental period. When fundamental power is nonzero, the composite wave equals the fundamental bandpass plus the second and third bandpasses scaled by the square roots of their power ratios to the fundamental.

An optional turning-point series

An optional rate-of-change of the composite wave, scaled by the fundamental period divided by 12.57, is described as crossing zero at cyclic turning points. One coded strategy buys the next bar at market when that rate-of-change crosses above zero and sells short the next bar at market when it crosses below zero.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
8 of 9 in the Bandpass filter track
20198-11 pp.Next on Bandpass filterLock a band, take a short lead, then gate empty readingsGroup delay is a lag in bars. Phase delay is that same lag written as a fraction of a cycle, so a fixed group delay is a larger phase shift on a shorter cycle.
All readings on this track · 9 readings
  1. 1994Constructing a cycle-aligned bandpass from paired lowpass filters
  2. 2008A bandpass filter bank for dominant cycle construction
  3. 2010Construct a bandpass, cycle, and trend mode detector
  4. 2015Constructing a one-parameter bandpass oscillator from two-bar momentum
  5. 2015Bandpass cycle models cannot promise certainty
  6. 2016Dual exponential Super Passband filter construction
  7. 2017A three-flag swing window with a bandpass midpoint
  8. 2019Building a three-harmonic Fourier series cycle wave
  9. 2019Lock a band, take a short lead, then gate empty readings
By fieldCycles and spectral analysis88 readingsBy year201992 readings
All 15 readings tagged Bandpass filter
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1984Constructing maximum-entropy spectra for dominant-cycle forecastsFourier analysis1986Why Fourier windows limit dominant-cycle resolutionFourier analysis1988Fourier cycle models break in major swingsFourier analysis1988Why a fitted dominant cycle is not a forecastFourier analysis1994Centering a dual lowpass bandpass on a counted cycle